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Theorem lubfun 18315
Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)
Hypothesis
Ref Expression
lubfun.u π‘ˆ = (lubβ€˜πΎ)
Assertion
Ref Expression
lubfun Fun π‘ˆ

Proof of Theorem lubfun
Dummy variables π‘₯ 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 6579 . . . 4 Fun (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))))
2 funres 6583 . . . 4 (Fun (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†’ Fun ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))}))
31, 2ax-mp 5 . . 3 Fun ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))})
4 eqid 2726 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 eqid 2726 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
6 lubfun.u . . . . 5 π‘ˆ = (lubβ€˜πΎ)
7 biid 261 . . . . 5 ((βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)) ↔ (βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))
8 id 22 . . . . 5 (𝐾 ∈ V β†’ 𝐾 ∈ V)
94, 5, 6, 7, 8lubfval 18313 . . . 4 (𝐾 ∈ V β†’ π‘ˆ = ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))}))
109funeqd 6563 . . 3 (𝐾 ∈ V β†’ (Fun π‘ˆ ↔ Fun ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))})))
113, 10mpbiri 258 . 2 (𝐾 ∈ V β†’ Fun π‘ˆ)
12 fun0 6606 . . 3 Fun βˆ…
13 fvprc 6876 . . . . 5 (Β¬ 𝐾 ∈ V β†’ (lubβ€˜πΎ) = βˆ…)
146, 13eqtrid 2778 . . . 4 (Β¬ 𝐾 ∈ V β†’ π‘ˆ = βˆ…)
1514funeqd 6563 . . 3 (Β¬ 𝐾 ∈ V β†’ (Fun π‘ˆ ↔ Fun βˆ…))
1612, 15mpbiri 258 . 2 (Β¬ 𝐾 ∈ V β†’ Fun π‘ˆ)
1711, 16pm2.61i 182 1 Fun π‘ˆ
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  βˆƒ!wreu 3368  Vcvv 3468  βˆ…c0 4317  π’« cpw 4597   class class class wbr 5141   ↦ cmpt 5224   β†Ύ cres 5671  Fun wfun 6530  β€˜cfv 6536  β„©crio 7359  Basecbs 17151  lecple 17211  lubclub 18272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-lub 18309
This theorem is referenced by:  joinfval  18336  joinfval2  18337
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